An Appraisal of the Ontario

Equivalent Base Length

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C. O'CONNOR

Research Report No. CEB

February, 1980

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CIVIL ENGINEERING RESEARCH REPORTS

This report is one of a continuing series of Research Reports published by the Department of Civil Engineering at the University of Queensland. This Department also publishes a continuing series of Bulletins. Lists of recently published titles in both of these series are provided inside the back cover of this report. Requests for copies of any of these documents should be addressed to the Departmental Secretary.

The interpretations and opinions expressed herein are solely those of the author(s). Considerable care has been taken to ensure the accuracy of the material presented. Nevertheless, responsibility for the use of this material rests with the user.

·

Department of Civil Engineering, University of Queensland, St Lucia, Q4067, Australia, [Tel:(07) 377-3342, Telex:UNIVQLD AA40315]

AN APPRAISAL OF THE ONTARIO EQUIVALENT

BASE LENGTH

by

C. O'Connor, DIC BD Land., BE PhD, MASCE, FIE AUST.

Professor of Civil Engineering

RESEARCH REPORT NO. CE 8 Department of Civil Engineering

University of Queensland

February, 1980

Synopsis

This report examines the equivalent base length

concept used for the development of the Ontario Highway

Bridge Design truck. As used in Ontario, the concept

does not rely on the assumption that a vehicle can be

described by two variables, the weight W and equivalent

base length EM' but rather that a vehicle can be

described by a series of points in W, EM space. This

report defines an alternative equivalent base, called

the concentrated base length, b, and introduces a

location parameter, x. The validity of the base length

concept is demonstrated by developing a vehicle

equivalent to a hypothetical family of Australian legal

vehicles. The accuracy of resultant curves of design

bending moment and shear is assessed. It is concluded

that reasonable accuracies can be achieved by the use of

a single, non-variable design truck, and that the Ontario

method is useful and important.

I '

CONTENTS

Page

1. INTRODUCTION 1

2. THE CONCEPT OF EQUIVALENT BASE LENGTH 1

3. COMPUTATION OF THE ONTARIO EQUIVALENT BASE 4 LENGTH

4. ALTERNATIVE DERIVATION OF EQUIVALENT BASE 5 LENGTH

4. 1 Typical Influence Lines 5

4. 2 Identification of the "Central" Load 6

4. 3 Ambiguous Case 8

4. 4 Concentrated Base Length 8

4. 5 Relationship between the Concentrated 10 Base Length and the Ontario Equivalent Base Length

5. VALIDITY OF THE EQUIVALENT BASE LENGTH 12

5. 1 Vehicles with the Same Base Length 12

5. 2 Vehicles with the Same Base Length and x 13

5. 3 Vehicles with Identical Envelopes of 15 Base Length Signature

5. 4 Conclusions

6. CHECK OF ONTARIO PROCEDURE

6. 1 Simulation Study

6. 2 Results of Simulation Study

7. CONCLUSIONS

8. ACKNOHLEDGEMENTS

APPENDIX A. NOMENCLATURE

APPENDIX B. REFERENCES

17

19

19

25

29

31

32

33

1. INTRODUCTION

1.

The 1979 Ontario Bridge Design Code (1) specifies

a design truck called the Ontario Highway Bridge Design

truck, or OHBD truck. The development of this idealized

truck was based on the concept of an Equivalent Base

Length (2, 3, 4) . This concept forms a major contribution

to the methodology of the selection of highway bridge

design loads, and could be used by other countries.

This paper offers an appraisal of the concept and

consists of (a) an alternative development of equivalent

base length, (b) a demonstration that vehicles of the

same total weight and equivalent base length may differ

in their effects on a bridge and (c) an examination of the

validity of the Ontario method as a whole by using it to

develop an equivalent vehicle from a series of Australian

legal trucks.

2. THE CONCEPT OF EQUIVALENT BASE LENGTH

The Ontario workers (4) have defined the

Equivalent Base Length, BM

, as "an imaginary finite length

on which the total weight of a given sequential set of

concentrated loads is uniformly distributed such that this

uniformly distributed load would cause force effects in a

supporting structure not deviating unreasonably from

those caused by the sequence itself".

The "sequential set of concentrated loads" may

correspond to a complete vehicle, or to any subset of

adjacent loads. If there are n loads in the vehicle as

a whole, then there are factorial n (n!) subsets of loads

taken 1, 2, 3 • . . n at a time. The total description of

the vehicle consists of n! points in (H, BM

) space,

where W is the total weight of the subset and BM

is the

equivalent base length. This series of points will be

referred to here as the equivalent base length signature.

2.

Consider any subset. Then the definition of

equivalent base length, as quoted above, implies that this

set of point loads may be replaced by a distributed load

w of length BM. It will be shown subsequently that this

description of the set by two values, W and BM

, is not

particularly good. However, the validity of this two­

variable vehicle description is not, in fact, crucial to

the equivalent base length concept, as used by the Ontario

Highway authorities. Rather, the concept may be describ ed

more fully as follows (3,4).

(1) Three surveys were carried out of truck traffic

in Ontario. Each truck, and each subset of axles,

were used to compute points (W, BM

).

(2) A histogram of such points was computed, of which

that shown in Figure 1 is an example (taken from

1967 vehicle survey data).

10 0 0

�· ,. 255 259 .. ""' • z

"

12 18 14 5 3 6 29 4 I

zo •• •• EQUIVALENT BASE LENGTH S... W FU.T

•• ••

FIGURE 1 : Histogram in vl/BM space, Ontario 1967 census data (4)

60 ..

(3) The line shown on the histogram of Figure 1 has the

following equation

vi (kips) � 20 + 2. 07BH

- 0. 0071B� (BM

in ft.) (1)

or

V1 (kN) 89.0 + 30. 2BM

- 0.340B� (BM

in m.) (2)

3.

This curve, which lies some distance below the

upper bound of the survey data, is called the

"Ontario Bridge Formula" (OBF).

(4) A subsequent 1971 survey was used to establish a

virtual upper bound of vehicles in Ontario. This

revised curve, called the Maximum Observed Load

(MOL), is given by

MOL (kips) 42. 5 + 2. 07

or

MOL (kN) 189. 0 + 30. 2

B -M

(for

B -M

0. 0071 B� ( 3)

BM

in feet)

0. 340 Bz M

( 4)

(for BM

in metres).

(5) The results of the 1967 and 1971 surveys were

confirmed by a third, 1975 survey.

(6) A vehicle was designed, the OHBD truck, with a

signature similar to the MOL curve (1,2,3,4).

This vehicle is shown in Figure 2.

z .X

I 1am I ��l���� 60 �� 7 2 §j Gross weight

I II l II(� SubtonfigUI'Qtions .J

BOO

600

·(

5 10 15 20 25 Bm (m)

FIGURE 2 OEBD truck (1,2,3,4)

4.

The crucial questions raised by the above

procedure are as follows.

(a) Is it valid to replace survey data by a histogram

of points in (H, BM

) space?

(b) Is it valid to select a design vehicle on the

basis that its equivalent base length signature

follows a curve such as the MOL curve?

3. COMPUTATION OF THE ONTARIO EQUIVALENT BASE LENGTH

The method used for computing BM

is described fully

in reference 4. It is based on the criterion that the

equivalent distributed load produce the correct ab solute

maximum bending moment in a simply supported beam.

It will be recalled that the maximum bending moment

under a particular load of a group which is moved across

the span occurs when the centre of the span bisects the

distance between that particular load and the centre of

gravity of the load group.

Consider a group, of total weight W, as shown in

Figure 3, and the bending moment under Pr

. Then, it may

EP; =W

d L

FIGURE 3 : Load group

be shown, (4), that a distributed load W of length BM

produces the same maximum bending moment if

4 z I Pi

xi I

w (5)

The span length, L, appears only in the second

term. This term is generally small. To avoid dependence

of BM

on L, reference 4 replaces equation (5) by

5.

(6)

The theorem quoted above does not identify the

load under which the absolute maximum bending moment occurs.

Reference 4 suggests that the load "closest to the centre

of gravity of the sequence" should be used.

4. ALTERNATIVE DERIVATION OF EQUIVALENT BASE LENGTH

4.1 Typical Influence Lines

In practice the design truck load may be used with

various influence lines to produce maximum values of bend­

ing moment, shear, reaction or axial force. The shape of

these influence lines is typically a series of straight

lines for a determinate structure, or a series of curves

for an indeterminate structure. The part of the influence

line that is often of greater importance - at least for the

longer spans - is near the largest ordinate.

REGION NEAR APPROXIMATION TO MAX. ORDINATE INFlUENCE LINES IN

Ff?'=l REGION NgAR MAX. ORDINATE

� (a) �I ��I I � (b) r==--I IF:il I

# I I� 1<=-1 (c) SL' , .

4--'===-' � (d)

FIGURE 4 : Some typical influence lines

Figure 4 shows some typical influence lines,

together with linear approximations in the region of the

maximum ordinate. Cases (b) - (d) may be modelled as a

pair of straight lines rising to the maximum at equal or

different slopes, as shown again in Figure 5. Such linear

6.

1 1 1 1 t,� 'T'

(a) (b)

FIGURE 5 : Common linear influence lines

influence lines occur commonly in determinate beams. In

particular, the influence line shown in Figure 5(b) is

the influence line for bending moment at the centre of a

simply supported span, and is of major importance in the

design of bridges of small span.

In the following sections it will be assumed that

the functions under consideration have influence lines

similar to Figure 5.

4.2 Identification of the "Central" Load

The maximum value of the function will occur when

one of the loads is at the maximum ordinate of the

influence line. This load will be referred to as the

"central" load. In many cases it is not clear which of

the loads is the central load, although for many influence

lines it will be a large load near the centre of the group.

Pc

I Total load =W

aPe

I Total load = W;z (0<a<1·0}

(1-alPc

I Total load ::W/2

FIGURE 6 Identification of "central" load

Figure 6 shows a load system, with resultant force

w. Define the central load Pc

as the load such that, when

aPe

, (O<a<l.O), is joined with the loads to the left, their

7.

resultant is !:!.2

. Similarly , (1-a) P with the loads to w c

the right also has resultant 2.

It will be shown that this definition of the

central load is correct for an equi- angular influence line

(i.e. as shown in Figure 4 (b)).

FIGURE 7

W,t.2:, P. Q c I ' I I

(1-alPc I LOAD SYSTEM

INFLUENCE LINE

Load system on equi-angular influence line

Let Pc

be located at the maximum ordinate, as

shown in Figure 7. Consider the change in the function F

as the complete load system is moved a distance s to the

right.

If the influence line were extended as shown dotted,

the change in F would be zero. The resultant � to the

left would rise through an ordinate ts on the influence

line, while the resultant to the right would fall an equal

amount.

More truly, F will reduce by an amount aPe

x 2ts,

where 2ts is the difference between the dotted extension

and the true influence line.

Suppose instead that the system is moved to the

left. Then, by a similar argument, F will reduce by

(1- a)Pc

x 2ts.

It follows that, provided O<a<l.O, the maximum

8.

value of F will occur with Pc

at the maximum ordinate.

It cannot be argued that the same load will be the

"central load" for an asy mmetric influence line. Rather

the validity of the above definition of the central load

will depend on the magnitude of a and the relative

magnitudes of the slopes tL

and tR

.

In the present argument, it is desirable to have

a unique definition of the central load. The definition

given above will be adopted as being correct for the

important case of an equi- angular influence line, and a

reasonable choice for many others in which the slope

difference is not great.

4.3 Ambiguous Case

The above definition of the central load is

ambiguous in the case where a = 0 (or 1.0). Pc

may be

identified by summing the loads from the left and stopping

when the sum is equal to or greater than w 2. If the sum

w equals 2, then either the current wheel, or that next to

the right, may be the central wheel.

In this case an integral subset of the loads w

happens to total 2. Such a case is shown in Figure 8.

W;z W;z mm

Q b

� 1 1 (a)EQUI·ANGULAR INFLUENCE LINE

W!z m

Q b

� 1 1 (b) UNSYMMETRICAL INFLUENCE LINE

FIGURE 8 : Ambiguous central load

Case (a) shows the loads applied to an equi-angular

influence line, with the ambiguous central loads straddling

the maximum. Then the function F is unchanged as the load

9.

system is moved from load b at the maximum to load a at

the maximum. That is, in this case, either a or b may be

used as the central load.

This situation does not occur in Case (b). Here

it is clear that for tL

> tR

' the load system must be

moved to the right until load a is at the maximum. In

practice, a design truck load system may be reversed.

With this in mind the worst central load may be defined as

that one of the loads a and b which is the smaller distance w

from the resultant of its 2 group.

4.4 Concentrated Base Length

rCE NTRAL LOAD

aP.

) ORIGINAL

LOAD

SYSTEM

Tl'

' ':_JP. ___ , -----'-" CENTRAL�

W POINT W V2J b b ' � EQUIVALENT CONCENTRATED

LOAD SYSTEM

L R

FIGURE 9 Concentrated base length

The above analysis suggests that the load system

shown in Figure 9 may be replaced by an equivalent load

system consisting of two concentrated loads ; located at

the resultants of the left and right halves of the load

sy stem. The distances, bL

and bR

, from the central load

to these resultants may be readily calculated.

Define the concentrated base length, b, as

b (7)

Define also the location parameter, x, as the b

L b

R lesser of band b x defines the "central point" of

the equivalent system.

10.

This equivalent load system, when applied to any

linear influence line, will produce the same value of the

function F as the original system, provided that Pc and

the central point are both placed at the maximum ordinate.

It is worth repeating this conclusion. The

equivalent load system may be used to find a maximum value

of the function F, with the load system facing in either

direction, and with the central point at the maximum

ordinate. Then this maximum will be the same as for the

original load system, provided that the worst load (i. e.

the one at the maximum ordinate) is indeed that defined

above as the "central load".

4. 5 Relationship between the Concentrated Base Length and the Ontario Equivalent Base Length

It was noted previously that in the definition of

the Ontario equivalent base length, equation ( 5) or (6) ,

the second term was small. Suppose it is neglected.

Assume also that the central load, Pr' is as defined above.

Then, for the left hand group of loads,

l: I I w P. x. 2

bL. l. l.

Similarly, for the right hand group,

l: I P. w x. 2

bR

. l. l.

For the complete load system,

4 l: w

2b.

That is, the simplified value of BM is precisely

equal to twice the concentrated base length, subject only

to the restriction that the same central load must be

used.

11.

There is value in the concept of an equivalent

distributed load. For example, it causes the equivalent

load to be spread over a realistic area of a complex

influence line. Furthermore, it is tempting to spread each

concentrated load ; over a length b, as shown in Figure 10,

to give a total length, 2b, of distributed load.

(a) EQUIVALENT CONCENTRATED LOAD

(b) EQUIVALENT DISTRIBUTED LOAD

FIGURE 10 : Relations hip between equivalent concentrated and distributed loads

This distributed load is not, however, precisely

equivalent to the system of two concentrated loads. The

distances bL

and bR

were originally calculated so that the w w

moments bL

x 2 and bR

x 2 equalled the moments about the

central wheel of the loads to the left and the right. It

may be observed that the two portions of the distributed

load do not satisfy this condition.

Furthermore, given the equivalent distributed load,

there is a temptation to assume that it may be placed any­

where on the influence line to achieve a maximum value of

the function F. It may be readily shown that this, in

general, leads to incorrect results.

For these reasons, it will be preferred in this

paper to use the equivalent concentrated load system defined

by the three variables, W, b and x. For the ambiguous case,

the central load will be taken as that wh ich gives the

smallest value of x. This is equivalent to the principle

stated previously.

12.

5 . VALIDITY O F THE EQUIVALENT BASE LENGTH

The equivalent loads described previously give

exact results only in some simple cases. It is desirable

to check the quality of the substitution for a range of

cases.

5. 1 Vehicles with the Same Base Length

Figure ll (a) shows a standard T44 truck. This is

the shortest form of a standard vehicle specified by the

NAASRA bridge code (5). It is based on the AASHO HS20

truck, increased by a factor 1. 349, and with double axles

(6) .

96196

1

tCENTRAL LOAD 96l 96l 4BkNI

(a) 11-21 3·0 11-2 1 37m I T44 x=0-3364

345·6, 86·4 kNj T1

(b) I 12·06m I X=O·O

216l

216kNI

(c) � T2 x=O·O

f9·6 259·2

1 43-2kN I

I 6·03 ·I 6·03m I T3 x=0·25

259·2'

129·6,

H2kNI

(d)

,J 4·82 I 4 · 82m I T4 x=O·O

1441 144kN I I

(e)

(f) I 3·62 .1 362m I T5 x=0·5

FIGURE ll Trucks with the same W and b (T44, Tl-T5)

13.

The complete T44 truck has a weight of 432 kN and

a concentrated base length b = 4. 82 m. Figure 11 also

shows five other trucks with the same total load, 432 kN,

and b = 4. 82 m. The locations of the central wheels, and

the values of x are shown.

Figure 12 (a) shows curves of maximum central

bending moment, for a simply supported girder plotted

against span. For the longer spans the curves are identical.

On the other hand, Figure 12 (b) shows similar

curves for support bending moment of a three span bridge

with cantilever and suspended spans. The horizontal

scale of the plot is deceptive in this case as L is the

total length rather than the length of any one span. Never­

theless, it may be seen that the curves differ, even for

the longer spans. Truck T3 has a value of the location

parameter x closest to that of the T44 truck and has given

the best results. All three trucks with x = 0 - i. e. Tl,

T2 and T4 - have given identical results for the longer

bridge lengths.

Similar curves have been obtained for a variety of

other cases - using influence lines for shear and bending

moment for determinate and continuous girders. Variations from

the T44 truck are similar in scale to those shown in Figure

12 (b) , although the T3 truck is not closest to the T44 truck

in all cases.

5 . 2 Vehicles with the Same Base Length and x

The vehicles shown in Figure 13 have the same

values of W, b and x. Figure 14 shows some results obtained

with these trucks.

In Figure 14 (a) , the central bending moments for a

simple span are consistent, except for the shorter spans.

For the support moment of a three span cantilever bridge,

Figure 14 (b) shows that after divergencies at the smaller

14.

(a) Central BM, simple span

-1 1-

� §i -1 1-� C)

&:­�

(b) Support Bt.l, 3 span cantilever bridge

FIGURE 12 : Curves of design bending moment, trucks T44 and Tl-T5

15.

T6

6·469

'"'I ";I· .. 54 k"l T7

�I �3�l�44�� -------1=2�·B=m�-------4· ' TB

FIGURE 13: Trucks with the same W, b and x (T6-T8)

spans, results converge for the longer spans. Figure

l4 (c) is for the shear at the end of a continuous girder,

and shows substantial differences for all spans.

5.3 Vehicles with Identical Envelopes of Base Length Signature

Figure 15 shows two trucks with identical envelopes

of base length signature. Subset l consists only of the

largest load. Subset 2 has the largest pair of adjacent

loads. Subset 3 contains all three loads. For the first

two subsets, x 0 for both vehicles. For the complete

vehicles, the values of x are 0.462 and 0.231 respectively.

For a 3- load system, there are 6 subsets of adjacent

loads. For truck T9, there are as follows

3 single loads

2 double loads

1 triple load

70, 100, 40 kN;

70 + 100, 100 + 40 kN;

70 + 100 + 40 kN.

The two load systems have identical signatures

except in regard to the smaller of the double load subset

- i.e. 100 + 40 for T9, and 40 + 70 for TlO. If the

/ /

!<·

t•

FIGURE 14

TB..- _.. ....

T6

(a) Central BH, simple span

(b) Support BM, 3 span cantilever bridge

(c) End shear, 3 span continuous girder

Curves of design bending moment and shear, trucks T6-T8

17.

signatures were truly identical, then the load systems as

a whole would become identical. Here the phrase has been

used because they have identical envelopes of base length

signature.

For these two trucks, agreement for the central

bending moment of a simple span was perfect for all spans.

70l

I 701

I

1001

4·0 I 70l

3-0 I Base

Subset 1 2 3

40kNl

6·0m I T9

100kNl

4-Dm I T10

length signature

w b 100 0·0 170 3·294 210 4·952

x=0·462

X:0·231

FIGURE 15 Trucks with identical envelopes of base length signatur� (T9, Tl0)

Figure 16 shows curves of bending moment at the

interior support for (a) a 3 span cantilever bridge and

(b) a 3 span continuous girder. In both cases significant

differences occur.

5. 4 Conclusions

The conclusions reached from the above studies

were as follows.

(1) No equivalent vehicle is perfect for all cases.

{2) Vehicles with identical envelopes of base length

signature can give different results.

(3) It is difficult to judge if the differences shown

to exist between hypothetical vehicles are

similar. in magnitude to those which may occur

� '

',

'

18.

' '

',

',·

'

T10' ',

',

, '

,,

' '

' '

' '

,, ' '

' '

' '

(a) Support BM, 3 span cantilever bridge

(b) Support BM, 3 span continuous girder

FIGURE 1 6 Curves of design bending moment, trucks T9, TlO

19.

with practical vehicles.

(4) There is some prospect of designing a satisfactory

equivalent vehicle by the use of the following

criteria -

(a) same maximum single axle load (or possibly

the same worst axle group);

(b) same envelope of base length signatures;

(c) for the total vehicle, the same total load,

concentrated base length, and location

parameter.

6. CHECK OF ONTARIO PROCEDURE

6. 1 Simulation Study

It was decided therefore to simulate the entire

Ontario process described in reference 4. The following

steps were taken.

(1) A series of vehicles was generated with configur­

ations lying at the Australian legal limits. These vehicles

are illustrated in Figure 17 and may be described as

follows.

(a) there are 2, 3 or 4 axle groups;

(b) the front group has 1 or 2 axles; the other

axle groups have 1, 2 or 3 axles, except that the second

group is not permitted to have fewer axles than the first,

and the fourth group, if it exists, has the same number of

axles as the third;

(c) the spacing of adjacent axles for a 2 axle

group is 1. 2 m, and for a 3 axle group 1.0 m;

(d) the spacing of adjacent axles of groups 1 and

2 is 3 rn, and either 3 or 6 m for other groups;

(e) vehicles longer than 16 m between extreme

axles are excluded;

(f) the �eight of an axle group is, where possible,

20.

Group 2 3

Numbi;l�I ti tr31 oy r i oy r I (al Ax!e 1,_21 lOrn 1-0 1{) 30or60ll0 !1 ol3{)or 6Dh ·0'1 0 conf1gurahon f.!4--' (b) Australian legal limits (maximum values l

1 axle group 8·5 t 2 axle group- steering 10 t 2 axle group-non-steering 1St 3 axle group 1Bt Maximum length 16m Maximum total weight varies from .1Bt at 3m total length to 36t at 12m,and then constant at 36t to 16 m.

FIGURE 17 : Family of Australian legal vehicles

set at the Australian legal limit, as shown in Figure 17;

but the total weight of the vehicle is, when necessary,

brought back to the overall legal limit (also shown in

Figure 17) ;

(g) in reducing a total vehicle to the legal

limit, two vehicles are generated - these having either

group 2 or 3 kept constant, and the remaining groups reduced

in the same proportion.

(2) For each vehicle, every subset of adjacent axles

was analysed to give W, b and x.

(3) Separate counts were made of the occurrences of

particular combinations of W and b, for x in the ranges

0-0. 1, 0. 1-0. 2, 0. 2 - 0. 3, 0. 3- 0. 4, 0. 4- 0. 5 , and for all x.

These counts were presented in the manner of Figure 1.

(4) Every vehicle was used to find maximum values for

the five functions shown in Figure 18, for L varied in

steps of 1 m over the range 1-100 m. The upper bounds or

envelopes of these functions were plotted against L.

(5 ) The W/b charts described in (3) were used to

generate equivalent trucks. These were used to obtain

separate curves of maxima of the five functions, for L

varying from 1-100 m, and these curves were compared with

A A F 1 �I __,L"----1•1

A A F2 I 0·75 L I

F3

,J � F4 I L I

21.

L

AI £> F5 � 0·75l .� l

Central bending moment

A A Central bending 1 0·75 L 1 moment

;{,\: A

Support bending moment

End shear

1 0·75 L 1 End shear

FIGURE 18 : Functions Fl - F5

the envelopes obtained in (4).

The programme, when used with the data described

above, generated 191 trucks, and 4050 subsets of axles,

although not all of these were dissimilar. The W/b chart

for all x is presented in Figure 19. The envelopes to this

chart, and to the charts for separate ranges of x, are

shown in Figure 20.

This figure also lists the coordinates of the

characteristic points of the upper envelope. These points

are recorded at the centre of the appropriate histogram

steps - W in steps of 10 kN, and b in steps of 0.25 m. The

subsets corresponding to some of these points may be

identified.

(a) The point (85' 0.125) is, more precisely, (83.4,

0) and corresponds to the legal single axle.

{ b) Similarly, point (145, 1.125) is (147.2, 1.2)' due

to the legal double axle, with axle spacing 1.2 m.

(c) Point (175, 1.375) is (176.6, 1. 33) due to the

I I L

22.

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ooooooooooooooooooooooooooooooo�:�tt:�oo

II :��������l���������l�l�l���l��lll��l��ll�� � ��=�•n•MNMO�m�<n•"N�O�m�·��"NMO�m��n•MN" -¢M�MMMMMM"MNNNNNNNNNNMMMMMMMMMM �

400

300

z ==-200 3

= 0·1-0·2 r----1 1 I

! I � I i :0·4-0·5

X= 0-0·1

1 2 3 4 5 6 7 b (m)

FIGURE 20 : Envelopes to H/b charts - Australian legal vehicles

8

"' w

24.

legal triaxle, with axle spacings 1. 0 m.

The nature of these points could be deduced with

reasonable confidence from the histogram itself. Three

equivalent legal trucks have been generated, as shown in

Figure 21.

I 58.1) 46·6 83·4 5rl

5r 14tkN ' W =353·2 kN b =5·875 m

I lwl,.ol2-39m 11·21 X :0·431 5·38 L T 1

I 46·6 58·9 83-4 4lu j

sy 15rkN W=353·2 kN l b =5·875m � j1-2j n9!1+nl X :0·1702

5·4m LT2

I 73-6 61-3

l61

61-3 83-4kN l l ' W=353-2kN

b =5·875m X :0·284

LT 3

FIGURE 21: Australian equivalent legal trucks (LT1-LT3)

(a) All contain one single axle at the legal limit of

B. St : 83. 4 kN.

(b) Trucks LTl and LT2 ·contain also a three axle group

at the legal limit (1St : 176.6 kN), together with a two

axle group chosen so that the total weight is at the legal

limit of 36t : 353. 2 kN.

(c) Truck LT3 has a group of two axles at the legal

limit (1St : 147.2 kN), as well as a legal single axle, and

then is made up of two additional equal single axles, each

being below the legal limit.

The signatures of those trucks all lie at the

25.

envelope for all x shown in Figure 20. This figure shows

that the envelopes of W tend to rise as x increases. That

is, there is a presumption that the vehicle which will best

simulate the worst legal truck will have a high value of x.

Figure 21 lists x for the three complete vehicles. LTl has

the largest x and on this principle should be preferred.

6. 2 Results of Simulation Study

The family of legal vehicles produced the curves

of the functions Fl-F5 against L shown in Figures 22 and

23. It may be noted that they are regular, even for small

spans.

9000 BODO

�7000 �

�6000 LU �5000 L:

�4000 D �3000

2000 1000

F1

20 30 40 50 60 70 80 90 100 L(m)

FIGURE 22 Curves of design bending moment Australian legal vehicles

340

300

260

-220 z ...lo::

0::: 180

<( �140 V)

10

FIGURE 23

20 30

26.

40 50 60 L (ml

70

F4

FS

BO 90 100

Curves of design shear - Australian legal vehicles

Figure 24 shows the departures from these curves

when the three equivalent vehicles are used. It is

important to compare the scales of these error curves with

the scales of the functions themselves, as shown in Figures

22 and 23 . The following general conclusions may be drawn.

(1) The errors are not large.

(2 ) Truck LTl tends to perform best, particularly at

the longer spans.

However, all curves show significant local errors

at small spans - particularly with shear. Table 1 lists

the errors at some significant points of the curves.

Calculations were carried out at 1 m intervals of the

length L. Slightly larger errors may be expected at inter­

mediate points.

"'

0

"'

"' ...

20 LT1.2,3

27.

0 I

----------------------------- LH e •\ � -: Lll \

e :Z .>::

-60 -80

20

\ L.-·-·--·-·-·-·-·-· -·-·-·-·-·-·--·-·--LT2

20 40 60 80 100

" "----

20 40 -6o--·-·---·-·--m 80 100

,---------------- --------LT3 I

LT1&2 I

\r----------LT1 V \....·-·-·--·---·-----·--· -·-·---·-LT2

-20 LT1,2&3 20 40 60 80 100

60 80 100

-12 0���2�0��40���60���80��100

L (m)

F1

F2

F3

F4

FS

FIGURE 24 Errors due to the use of the equivalent trucks LT1-LT3

TABLE 1 Errors using the Equivalent Legal Trucks LT1-LT3

!Function Truck

LT1 LT2 LT3

L Error % error L Error % error L Error F1 9 m +11. 6 kNm +3.2 9 m +11. 6 kNm . +3.1 4 m - 14.7 kNm

(BM) 13 - 12.9 - 2.0 18 - 87.4 - 8.1 9 +11. 4 16 + 4.4 +0.5 100 - 87.0 - 1.0 14 - 18.4

100 + 5.4 +0.06 100 - 2.8

F2 6 m - 14.9 kNm - 12.1 5 m - 9.8 kNm - 10.2 5 m - 13.9 kNm

(BM) 9 - 15.2 - 6.9 8 + l. 5 + 0.8 7 - 4 •. 0 14 + l. 7 + 0.4 11 - 19.6 - 6.5 10 ' - 13.8

100 + 4.7 + 0.08 100 - 78.0 - l. 45 100 - 3.0

F3 4.

m - 14.7 kNm - 14.3 4 m - 14.7 kNm - 14.3 7 m - 3.3 kNm

(BM) 8 +14.3 + 5.4 8 +14.3 + 5.4 8 4.7 15 +25.3 + 3.5 15 +25.3 + 3.5 10 - 12.2

100 +11. 7 + 0.14 20 - 7.3 - 0.7 21 +49.9 100 - 5.0 - 0.06 100 +50.7

F4 2 m - 14.7 kN - 14.3 2 m - 14.7 kN - 14.3 5 m + 6.7 kN (Shear) 5 + 4.1 + 2.9 5 + 4.1 + 2.9 8 - 11.0

8 + 5.7 + 3.1 8 + 5.7 + 3.1 29 + 9.9 14 - 9.3 - 3.9 28 +13.1 + 4.6 100 + 2.9 28 + 5.2 + 1. 8 100 + 3.7 + 1.10

100 + l. 5 + 0.44

F5 3 m - 11.1 kN - 10.9 9 m - 6.4 kN - 4.0 3 m - 7.3 kN (Shear) 12 + 5.5 + 3.1 15 + 5.0 + 2.5 7 + 5.1

21 - 9.8 - 4.2 45 +13.0 + 4.6 12 - 10.4 45 + 4.9 + 1. 7 100 + 6.0 + l. 87 45 +10.0

100 + 2.4 + 0.74 100 + 4.7

% error - 12.5 + 3.1 - 2.5 - 0.03

- 14.5 - 2.7 - 5.3 - o. 05

- 1. 5 + 1.8 - 3.1 + 4.3 + 0.62

+ 4.7 - 6.0 + 3.5 + 0.87

- 7.2 + 3.7 - 5.8 + 3.5 + l. 46

N CX)

29,

Table 1 shows that truck LTl generally gives the

best results, particularly for the longer spans. For

all trucks, significant errors occur at small spans - of

the order of 10- 15% for L equal to 6 m or less. These

errors clearly result from the difficulty of simulating

all axle groups accurately as well as the vehicle as a

whole.

The Australian legal limits have the characteristic

that the restriction on total weight makes it unlikely that

all axle groups will be fully loaded. For example, a

vehicle with three axle groups, with 1, 2 and 2 axles

respectively in the groups, has a sum of legal axle group

loading of 38.5 t, which already exceeds the limit on total

weight (36t). At the upper bound, the sum of the legal

axle group weights for 2, 3, 3 and 3 axles would be 64t,

greatly in excess of the 36t limit.

It is impossible to incorporate in a single vehicle,

the legal axle group weights for the 3 possible groups,

with l, 2 and 3 axles in the groups.

In view of this, it is considered that the results

are surprisingly good. It is difficult to see how the

equivalent trucks LT1-LT3 could otherwise have been derived.

7. CONCLUSIONS

(1) The concept of equivalent base length (BM

or b) is

extremely useful in the determination of an appropriate

highway bridge design truck.

(2) It needs, however, to be considered with care. It

is impossible to describe a complex truck accurately by two

values only - H and BM

' or H and b - nor is this implied in

the use made by the Ontario authorities of the equivalent

base length concept. Rather, it is implied that any single

vehicle may be described with reasonable accuracy by its

signature in {l'i', BM

) or (;'l, b) space.

30.

(3) This paper develops an equivalent concentrated

base length (b) . Although this differs from the Ontario

distributed base length (BM) , these differences do not

have significant practical consequences. The concentrated

base length is precisely 0. 5 x an approximate form of the

Ontario Equivalent Base Length.

( 4) The description of a subset of axles may be

enlarged to include a location parameter, x, and this may

be used to choose between equivalent vehicles with similar

base length signatures.

( 5) A study using a family of hypothetical Australian

legal vehicles shows that the Ontario method may be used

to choose equivalent vehicles which give accurate curves

against span of the maxima of five functions - (a) bending

moment at mid-span of a simply-supported girder, (b) bending

moment at mid-span of a 3-span continuous girder, (c) bend­

ing moment at the support of a 3-span cantilever/suspended

span girder, (d) shear at the end of a simple girder, and

(e) shear at the end of a continuous girder. It may be

expected that similar accuracies would be achieved in other

cases.

(6) This study should also assist the selection of

an improved Australian design vehicle. It is proposed to

extend the study to investigate the effects of possible

changes in the Australian legal limits - particularly in

the specification of total mass or weight.

(7) The present AASHO design truck has a variable back

axle spacing. With the new OHBD truck, the designer is

required to consider various subsets of axles. The present

study suggests that it may be possible to use a single,

non-variable vehicle with sufficient accuracy. This would

lead to a reduction in the time involved in the computation

of design parameters, such as bending moment and shear.

This may be of particular value in computer-aided design,

where the use of variable vehicles can add significantly

to logical complexity and computation time.

8 • ACKNO\�LEDGEMENTS

31.

This study has arisen from work which has been

carried out at the University of Queensland on project

P321 of the Australian Road Research Board. This is part

of a more general study of truck loads in Australia.

Some of the computations were performed by Neville Richter,

Tutor in Civil Engineering. The paper has been typed by

Eileen Hoare.

l_

APPENDIX A

a

b

n

OBF

p c

s

t

32.

NOMENCLATURE

fraction of central load

concentrated base length

equivalent lever arms - left, right

Ontario equivalent base length

length of load group

functions, such as bending moment and shear

span or characteristic length

maximum observed load

number of loads

Ontario bridge formula

concentrated loads

"central" load

movement of load group

slope of linear influence line

slopes at left and right of linear influence line

location parameter

distance to load Pi

from reference load Pr

total load

33.

APPENDIX B REFERENCES

1. Ontario Highway Bridge Design Code 1979, Ontario Ministry of Transportation and Communications, Jan. 1979.

2. Ontario Highway Bridge Design Code 1979, Vol. II, Ontario Ministry of Transportation and Communications.

3. CSAGOLY, P.F., DORTON, R.A., "The Development of the Ontario Bridge Code", Ontario Ministry of Transportation and Communications, Oct. 1977.

4. CSAGOLY, P.F., DORTON, R.A., " Truck Weights and Bridge Design Loads in Canada", paper presented at the AASHTO Annual· Meeting, Louisville, Kentucky, Nov. 1, 1978.

5. NAASRA Bridge Design Specification, National Association of Australian State Road Authorities, Sydney, 1976.

6. American Association of State Highway Officials, Standard Specifications for Highway Bridges, 11th ed., 1973.

CE No.

l.

2.

3.

4.

5.

6.

7.

8.

CIVIL ENGINEERING RESEARCH REPORTS

Title

Flood Frequency Analysis: Logistic Method

for Incorporating Probable Maximum Flood

Adj ustment of Phreatic Line in Seepage

Analysis by Finite Element Method

Creep Buckling of Reinforced Concrete

Columns

Buckling Properties of Monosyrnrnetric

!-Beams

Elasto-Plastic Analysis of Cable Net

Structures

A Critical State Soil Model for Cyclic

IDading

Resistance to Flow in Irregular Channels

An Appraisal of the Ontario Equivalent

Base Length

Author(s)

BRADY, D.K.

ISAACS, L.T.

BEHAN, J.E. &

O'CONNOR, C.

Date

February,

1979

March,

1979

April,

1979

KITIPORNCHAI, S. May,

& TRAHAIR, N.S. 1979

MEEK, J .L. &

BROWN, P.L.D.

CARTER, J.P.,

BOOKER, J.R. &

WROTH, c·.P.

November,

1979

December,

1979

KAZEMIPOUR, A.K. February,

& APELT, C.J. 1980

O'CONNOR, C. February,

1980

CURRENT CIVIL ENGINEERING BULLETINS

4 Brittle Fracture of Steel - Perform­

ance of ND 18 and SAA A 1 structural

steels: C. O'Connor ( 1964)

5 Buckling in Steel Structures - 1. The

use of a characteristic imperfect shape

and its application to the buckling of

an isolated column: C. O'Connor

( 1965)

6 Buckling in Steel Structures - 2. The

use of a characteristic imperfect shape

in the design of determinate plane

trusses against buckling in their plane:

C. O'Connor ( 1965)

7 Wave Generated Currents - Some

observations made in fixed bed hy­

draulic models: M.R. Gourlay ( 1965)

8 Brittle Fracture of Steel - 2. Theoret­

ical stress distributions in a partially

yielded, non-uniform, polycrystalline

material: C. O'Connor ( 1966)

9 Analysis by Computer - Programmes

for frame and grid structures: J.L.

Meek ( 1967)

10 Force Analysis of Fixed Support Rigid

Frames: J.L. Meek and R. Owen

( 1968)

1 1 Analysis by Computer -- Axisy· metric solution of elasto-plastic pro­blems by finite element methods: J.L. Meek and G. Carey ( 1969)

12 Ground Water Hydrology: J.R. Watkins

( 1969)

13 Land use prediction in transportation planning: S. Golding and K.B. David­son ( 1969)

14 Finite Element Methods Two dimensional seepage with a free sur­face: L. T. Isaacs ( 197 1)

15 Transportation Gravity Models: A. T.C.

Philbrick ( 197 1)

16 Wave Climate at Moffat Beach: M.R.

Gourlay ( 1973)

17 Quantitative Evaluation of Traffic

Assignment Methods: C. Lucas and K.B. Davidson ( 1974)

18 Planning and Evaluation of a High Speed Brisbane-Gold Coast Rail Link:

K.B. Davidson, et at. ( 1974)

19 Brisbane Airport Development Flood­

way Studies: C.J. Apelt ( 1977)

20 Numbers of Engineering Graduates in

Queensland: C. O'Connor ( 1977)